Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,17,28)(8,12)(9,13)(10,14)(11,18,24,21)(16,32,33,31)(19,26)(20,27)(22,29)(23,30)(25,34,36,35);(2,4)(11,24)(15,28)(25,36)(31,32);(5,7)(6,11)(8,16)(9,21)(10,17)(12,25)(13,28)(14,24)(15,30)(18,23)(19,35)(20,33)(22,34)(26,31)(27,36)(29,32);(2,4)(5,8,10,20)(6,12,14,27)(7,16,17,33)(9,22,23,19)(11,25,24,36)(13,29,30,26)(15,31,28,32)(18,35,21,34)Polycyclic presentation:
2^9|a^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72302285, #72302882, #72302893, … |
| maximal normal subgroup of | #72302285, #72302882, #72302893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266755);